Module 5: Randomized Complete Block Designs (RCBD)

Analyzing a RCBD

What changes from a CRD?

If we analyze ignoring the effect of oven…

Incorrect Analysis: CRD

\(y_{ij} = \mu + \tau_i + \epsilon_{ij} \text{ where } \epsilon_{ij} \text{iid}\sim N(0,\sigma^2)\)

oven_crd_mod <- lm(texture ~ recipe, data = cookie_data)
anova(oven_crd_mod)
Analysis of Variance Table

Response: texture
          Df Sum Sq Mean Sq F value Pr(>F)
recipe     2  63.82  31.910  0.3459 0.7166
Residuals  9 830.24  92.248

Statistical Effects Model

\[y_{ij} = \mu + \tau_i + \rho_j + \epsilon_{ij} \text{ where } \epsilon_{ij} \text{iid}\sim N(0,\sigma^2)\] for \(i = 1, 2, 3\) and \(j = 1, 2, 3, 4\)

where:

  • \(y_{ij}\) is the observed texture of the cookie tray baked in the \(j^{th}\) oven with the \(i^{th}\) recipe.
  • \(\mu\) is the overall mean texture
  • \(\tau_i\) is the effect of the \(i^{th}\) recipe
  • \(\rho_j\) is the effect of the \(j^{th}\) oven
  • \(\epsilon_{ij}\) is the experimental error associated with the cookie tray baked in the \(j^{th}\) oven with the \(i^{th}\) recipe.

What Blocking does to the variability

\[SST = SSTrt + SSBlk + SSE\]

  • \(SST = \sum_i\sum_j(y_{ij}-\bar y_{\cdot\cdot})^2\)
  • \(SSTrt = r \sum_i(\bar y_{i\cdot}-\bar y_{\cdot\cdot})^2\)
  • \(SSBlk = t\sum_j(\bar y_{\cdot j}-\bar y_{\cdot\cdot})^2\)
  • \(SSE = t\sum_j(y_{ij}-\bar y_{i\cdot} - \bar y_{\cdot j} + \bar y_{\cdot\cdot})^2\)

ANOVA Table

Source of Variation DF SS MS F
Block r - 1 SSBlk MSBlk MSBlk/MSE
Treatment t - 1 SSTrt MSTrt MSTrt/MSE
Block x Treatment \(\rightarrow\) error (r - 1)(t - 1) SSE MSE \(\rightarrow \sigma^2\)
Total (N = rt) N - 1 SST

Blocking is a design tool, not usually a research question.

Skeleton ANOVA

Source of Variation DF

R: RCBD Analysis

oven_rcbd_mod <- lm(texture ~ recipe + oven, data = cookie_data)
anova(oven_rcbd_mod)
Analysis of Variance Table

Response: texture
          Df Sum Sq Mean Sq F value    Pr(>F)    
recipe     2  63.82  31.910  38.607 0.0003749 ***
oven       3 825.28 275.092 332.829 4.651e-07 ***
Residuals  6   4.96   0.827                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
emmip(oven_rcbd_mod, ~ recipe, CI = T)

emmeans(oven_rcbd_mod, ~ recipe) |> 
  cld(Letters = LETTERS, decreasing = T, adjust = "tukey")
 recipe         emmean    SE df lower.CL upper.CL .group
 Brown Butter     73.9 0.455  6     72.4     75.4  A    
 Classic Butter   69.5 0.455  6     68.0     71.0   B   
 Coconut Oil      68.7 0.455  6     67.2     70.2   B   

Results are averaged over the levels of: oven 
Confidence level used: 0.95 
Conf-level adjustment: sidak method for 3 estimates 
P value adjustment: tukey method for comparing a family of 3 estimates 
significance level used: alpha = 0.05 
NOTE: If two or more means share the same grouping symbol,
      then we cannot show them to be different.
      But we also did not show them to be the same.

JMP: RCBD Analysis

Check Model Assumptions – \(\epsilon_{ij} \text{iid} \sim N(0,\sigma^2)\)

par(mfrow = c(2,2))
plot(oven_rcbd_mod)

RCBD vs CRD

If we ignore ovens (CRD):

  • Oven variability inflates MSE
  • Harder to detect recipe differences
Analysis of Variance Table

Response: texture
          Df Sum Sq Mean Sq F value Pr(>F)
recipe     2  63.82  31.910  0.3459 0.7166
Residuals  9 830.24  92.248

With RCBD:

  • Oven variability is removed from error
  • Smaller MSE
  • More power (if blocks are different)
Analysis of Variance Table

Response: texture
          Df Sum Sq Mean Sq F value    Pr(>F)    
recipe     2  63.82  31.910  38.607 0.0003749 ***
oven       3 825.28 275.092 332.829 4.651e-07 ***
Residuals  6   4.96   0.827                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Why is this model additive? (no interaction)

  • Each recipe appears once per oven
  • No replication within recipe-oven combinations
  • Interaction is not estimable

Recall: The experimental error comes from the block x treatment term.